# Mutual information and bivariate function of independent variables

Let $$X, Y, Z$$ be discrete random variables with $$X$$ and $$Y$$ independent of $$Z$$, while $$X$$ and $$Y$$ can be dependent. For the mutual information, we have $$I(X; Y,Z) = I(X;Y)$$. Now consider $$I(X; f(Y,Z))$$ for some deterministic function $$f$$. Does $$I(X; f(Y,Z))$$ depend on $$Z$$? If not, is there a way to express $$I(X; f(Y,Z))$$ in terms of $$X$$ and $$Y$$ only?

$$I(X;f(Y, Z))$$ can depend on the $$Z$$ (or more specifically the distribution of $$Z$$). Consider the following example, $$Z \sim B(p)$$, $$X \sim B(0.5)$$, $$Y=X$$ satisfying $$X,Y \perp \!\!\! \perp Z$$. Let $$F=\max(Y, Z)$$ be a variable from the output of a deterministic function of $$Y, Z$$. $$X, F$$ have the following joint distribution,
1. $$P(X=0,F=0) = (1-p)/2$$.
2. $$P(X=0,F=1) = p/2$$.
3. $$P(X=1,F=0) = 0$$.
4. $$P(X=1,F=1) = 1/2$$.
$$I(X;F) = \log(2) + \frac{p}{2}\log(p) - \frac{1+p}{2}\log(1+p)$$ which is dependent on $$p$$.